Calculate the area of an irregular plot using the Trapezoidal Rule, given the offsets are 2 m, 4 m, 3 m, and 5 m at an equal spacing of 1 m.
$10.5 m^2$
To calculate the area of an irregular plot using the Trapezoidal Rule, we first need to understand the formula for this method of numerical integration. The Trapezoidal Rule is used to estimate the area under a curve by dividing the total area into trapezoids rather than rectangles.
The formula for calculating the area using the Trapezoidal Rule is:
\(A = \frac{d}{2} \times [(f_0 + f_n) + 2(f_1 + f_2 + \ldots + f_{n-1})]\)
where:
- \(A\) is the total estimated area.
- \(d\) is the distance between each set of offsets (in this case, 1 m).
- \(f_0, f_1, \ldots , f_n\) are the offsets given (2 m, 4 m, 3 m, and 5 m).
Given the offsets are 2 m, 4 m, 3 m, and 5 m, and the spacing \(d\) is 1 m, let's apply the formula:
- Identify the values: \(f_0 = 2\), \(f_1 = 4\), \(f_2 = 3\), and \(f_3 = 5\).
- Substitute into the formula:
\(A = \frac{1}{2} \times [(2 + 5) + 2(4 + 3)]\)
\(A = \frac{1}{2} \times [7 + 2 \times 7]\)
\(A = \frac{1}{2} \times [7 + 14]\)
\(A = \frac{1}{2} \times 21\)
\(A = 10.5 \, m^2\)
Thus, the estimated area of the irregular plot using the Trapezoidal Rule is \(10.5 \, m^2\).
The correct answer is $10.5 \, m^2$.
Similar Questions
A steel plate is 20 mm thick and 45 mm wide. What is the length of the diagonal across the plate?
In a workshop calculation, you need to find the side of a square base with an area of 64 cm2. What is the correct length?
Find the circumference of a circle having radius 7 cm. Use π=22/7.
- The volume of a solid cylinder is 5,236 cm$^3$ and its height is 34 cm. What is the total surface area of the solid cylinder? $\left(\text{Use } \pi = \frac{22}{7}\right)$
In figure 'o' is the centre of a circle. The area of sector OAPB is $\frac{5}{18}$ of the area of the circle find $x$.
Important Questions from Mensuration
In a circular garden of radius 15 m, a path of 2 m wide has to be made inside the garden at the rate of ₹ 24 per sq. m. The cost of making the path is: (Take π = \(\frac{22}{7}\) )
A hollow cylinder with outer radius 4 cm and height 2 cm is made up of 1 cm thick metal sheet. What is the volume of metal used? (Take π = \(\frac{22}{7}\))
The sides of a triangular park are 60 m, 297 m and 303 m. Its area is equal to the area of a square-shaped garden. What is the perimeter (in m) of the garden?
The length and breadth of a rectangular field are in the ratio 4 : 3. If the cost of cultivating the field at ₹2 per m 2is ₹600, then the length of the field is:
A cylindrical tank has a capacity of 5632 m3. If the diameter of its base is 8 m, what is the depth of the cylindrical tank? (Use π = \(\frac{22}{7}\))
